section 4.3 fundamental theorem of calculus math 1231: single-variable calculus
TRANSCRIPT
![Page 1: Section 4.3 Fundamental Theorem of Calculus Math 1231: Single-Variable Calculus](https://reader035.vdocuments.net/reader035/viewer/2022062300/56649dd25503460f94ac8283/html5/thumbnails/1.jpg)
Section 4.3 Fundamental Theorem of Calculus
Math 1231: Single-Variable Calculus
![Page 2: Section 4.3 Fundamental Theorem of Calculus Math 1231: Single-Variable Calculus](https://reader035.vdocuments.net/reader035/viewer/2022062300/56649dd25503460f94ac8283/html5/thumbnails/2.jpg)
(Signed) Area function
![Page 3: Section 4.3 Fundamental Theorem of Calculus Math 1231: Single-Variable Calculus](https://reader035.vdocuments.net/reader035/viewer/2022062300/56649dd25503460f94ac8283/html5/thumbnails/3.jpg)
Signed area function: example
Example If f is a function whose graph is shown in the Figure and , find the values of g(0), g(1), g(2), g(3), g(4), g(5).
![Page 4: Section 4.3 Fundamental Theorem of Calculus Math 1231: Single-Variable Calculus](https://reader035.vdocuments.net/reader035/viewer/2022062300/56649dd25503460f94ac8283/html5/thumbnails/4.jpg)
Signed area function: example
![Page 5: Section 4.3 Fundamental Theorem of Calculus Math 1231: Single-Variable Calculus](https://reader035.vdocuments.net/reader035/viewer/2022062300/56649dd25503460f94ac8283/html5/thumbnails/5.jpg)
Fundamental Theorem of Calculus: Part I
The signed area function defined by f is an anti-derivative of f.
![Page 6: Section 4.3 Fundamental Theorem of Calculus Math 1231: Single-Variable Calculus](https://reader035.vdocuments.net/reader035/viewer/2022062300/56649dd25503460f94ac8283/html5/thumbnails/6.jpg)
Examples
![Page 7: Section 4.3 Fundamental Theorem of Calculus Math 1231: Single-Variable Calculus](https://reader035.vdocuments.net/reader035/viewer/2022062300/56649dd25503460f94ac8283/html5/thumbnails/7.jpg)
Fundamental Theorem of Calculus: Part II
The Fundamental Theorem of Calculus, Part 2 If f is continuous on [a, b], then
where F is any anti-derivative of f .
Property
![Page 8: Section 4.3 Fundamental Theorem of Calculus Math 1231: Single-Variable Calculus](https://reader035.vdocuments.net/reader035/viewer/2022062300/56649dd25503460f94ac8283/html5/thumbnails/8.jpg)
Examples
![Page 9: Section 4.3 Fundamental Theorem of Calculus Math 1231: Single-Variable Calculus](https://reader035.vdocuments.net/reader035/viewer/2022062300/56649dd25503460f94ac8283/html5/thumbnails/9.jpg)
Differentiation and Integration: Inverse processes
The Fundamental Theorem of Calculus If f is continuous on [a, b], then
1. : If f is integrated and then is differentiated, we arrive back to f.
2. : If we take a function F, first differentiate it then integrate it, we arrive back to F, but in the form of F(b) – F(a).
![Page 10: Section 4.3 Fundamental Theorem of Calculus Math 1231: Single-Variable Calculus](https://reader035.vdocuments.net/reader035/viewer/2022062300/56649dd25503460f94ac8283/html5/thumbnails/10.jpg)
Importance of FTC
![Page 11: Section 4.3 Fundamental Theorem of Calculus Math 1231: Single-Variable Calculus](https://reader035.vdocuments.net/reader035/viewer/2022062300/56649dd25503460f94ac8283/html5/thumbnails/11.jpg)
Examples
![Page 12: Section 4.3 Fundamental Theorem of Calculus Math 1231: Single-Variable Calculus](https://reader035.vdocuments.net/reader035/viewer/2022062300/56649dd25503460f94ac8283/html5/thumbnails/12.jpg)
More Examples
![Page 13: Section 4.3 Fundamental Theorem of Calculus Math 1231: Single-Variable Calculus](https://reader035.vdocuments.net/reader035/viewer/2022062300/56649dd25503460f94ac8283/html5/thumbnails/13.jpg)
More Examples